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The Second Law Strikes Back

By Pradeep Mutalik March 5, 2010 11:12 amMarch 5, 2010 11:12 am

Last week we discussed the papaya battery, an utterly simple device whose geometry apparently creates a temperature difference where there was none before, thus violating the Second Law of Thermodynamics and creating potentially unlimited energy. In the field of thermodynamics, which abounds in apparent paradoxes, this is one of the most delicious paradoxes ever. Devices like the papaya battery, based on ellipsoids and parts of spheres, have been known to science since at least 1959 (J.C. Fallows in “New Scientist”) and have been discussed in the scientific literature in cutting-edge journals like the “Journal of Statistical Physics” and “Physics Bulletin” under the names “Ellipsoid Paradox” and “Chinese furnace” which interested readers can google.

In case anyone was misled by my tongue-in-cheek statements in last week’s blog, I must hasten to say that my brother Madhav and I were not the first ones to think up this paradox – we independently reconstructed it from a passing mention in a Scientific American article on entropy many years ago. Our contributions were in giving it its fruity name and we had fun coming up with compelling variations of it involving radial radiators and differently constructed bodies, which my brother demonstrated last week.

Of course, it doesn’t work, but the reason is not easy to see at first sight. In fact the papaya fails not because of any arcane physics but simply because of geometry: it fails because of the geometric equivalent of the dreaded division by zero.

There are many algebraic paradoxes which “prove” absurdities like 1=2, where there is a hidden division by zero. You can legally divide by any number, no matter how small it is, as long as it is not zero. But you can never divide by zero. Between zero and the smallest number you can think of is an unbridgeable gulf, a discontinuity for the operation of division.

In a similar way, the papaya battery disregards a geometric discontinuity. Idealized rays from one mathematical point focus of an ellipsoid do indeed go to the other focus. But, as was first pointed out in the discussion by Bart, a real-world physical body has a surface whose points cannot all be at the mathematical focus. And the rays from those points miss the other focus by a great deal. It is tempting to think that by making the body very small relative to the ellipsoid, we can make the rays go arbitrarily close to the other focus. Just like dividing by zero, that is not the case. Between a point and a surface is an unbridgeable discontinuity which is not obvious. (We saw this mental trap clearly in last week’s discussion when Arlin, who was on the right track and had previously contributed the valuable ideas of view factors and reciprocity took a small stutter-step in response to this question posed by Hans. However Arlin and Hans resolved this and triumphantly solved the problem in concert. Congratulations!)

The above diagram shows the fraction of the B-radiation reflected at five different points that goes on to strike body A for the ellipsoid we postulated. Note that the diagram is independent of scale. The bodies and ellipsoid could be many kilometers wide or merely nanometers: the fractions remain exactly the same. Consider point R, which is vertically above body B. As can be seen, all rays from body B that reach this point run through the triangle shown and are reflected towards A, forming another triangle which is geometrically similar. The ratio of the distances BR to AR is about 3:10. In three dimensions, this same ratio is rotated all around (body A presents an area that the rays must strike), so the actual fraction of reflected radiation from R that strikes A is the square of this ratio: 0.09 or only 9%. Once again, this is true no matter how small you make the bodies relative to the ellipsoid. (Of course, once the size of the bodies approaches a size comparable to the wavelength of the radiation then the rays go quantum crazy, and in the ensuing gaiety the rays can longer be trusted to be straight.)

To the right of body B, no rays that strike T are reflected to A. At point S, halfway between R and T, the fraction is only 3%, and the average for the whole segment RT (and its identical lower half) is 4%. Though fully half the radiation emitted by B is towards the right, only 4% of this half or about 2% of the total lands on A: 48% does not. One of the seductive properties of the papaya battery is that it is easy to see that 50% of all rays emitted by A go to the hemisphere and return to A. Surely, you think, with such a big “lead” in rays, B cannot catch up. But A pays a price for the perfect reflection of the hemisphere: the flip side is that any ray that does not originate from A cannot end up on A on reflection by the hemisphere. So all the 48% of rays that start out on right side of B and miss A are reflected back by the hemisphere, where they again definitely miss A, and so are much more likely to end up on B rather than A. The same is true of rays emitted to the left between points Q and R. Of this bunch, which constitute 25% of the radiation from B, only a total of 9% hit A. The remaining 16% are reflected back by the hemisphere, and are again much more likely to end up on B. If you add to this the 10% of rays that directly strike the hemisphere from B, that’s 74% of rays from B that mostly end up back on B. This compensates for A’s hemispheric advantage, and if you do a computer simulation and tally all the rays, they end up equal: about 100 on A and about 100 on B, with very small random statistical fluctuations. Look ma, no temperature difference!

What about the case of radial radiation? Surely, now the rays are coming from the focus and will end up on A? Again the answer is no, for the same reason. The atoms on the surface of the bodies that are doing the radiating are finite in size, and the rays can start anywhere on their extent. The rays are therefore slightly deviated from the mathematical line going through the focus. This deviation is magnified on reflection, and they land on A at an angle that is not radial, so they bounce off, are reflected back from the hemisphere, bounce off A again and end up back on B.

But wait a minute, you say. Surely the thermodynamic cop, the Second Law (to adopt the image from Cuda Baloo’s well-recommended and accurate poem) cannot police every possible geometric configuration that might be suggested by someone’s inventive brother in this ad hoc way. How does the Law guarantee that all such ingenious arrangements will fail? Surely it has some deeper trick up its sleeve. Yes, it does! The simple trick is reversibility: the Second Law works because every path that a ray (or an object) can take is reversible. If B radiates along a path to A, no matter how complicated, then A can radiate along the same path back to B. This fact was used by Hans to come up with the general proof that no such arrangement will work. It is a remarkable paradox that the reversibility of the Laws of Physics is indeed the secret weapon of the Second Law, which is the only thing in the universe which is irreversible! In an almost mystical process, the Second Law takes reversibility, and by the power of the statistics of large numbers, turns it into irreversibility, so that the arrow of Time always clearly moves in the forward direction in non-quantum situations.

To understand Hans’s deep proof and the concepts of the view factor and reciprocity that Arlin introduced, I’d like to take you through a simple thought experiment. It uses an innovation that heads my list of the world’s all-time favorite unessential inventions: the bendy straw (fitting, since we are talking about a fruit).

The second figure above shows six colored bendy straws following paths light rays can take from A to B and back. Now I invite you to exercise your imagination. Imagine that these bendy straws are more sophisticated than actual bendy straws: they can be as long as you want, and they can bend whenever they encounter the inner surface and can do this any number of times. They can pass through each other without interfering. Their width is uniform throughout, but they can be so thin that for every radiating atom, there can be a separate straw for every direction that it can radiate. These are all known characteristics of light radiation (photons).

Now mentally place the bendy straws to trace every path that goes from A to B. Some of these might have single reflections like the ones shown. Others might have two, or three or a thousand. Never mind, our magic mental bendy straws will cover them. You don’t actually have to know all the paths, just imagine they are all covered. Now by definition, any other ray leaving A that doesn’t go through the straws can only land back on A, and any ray leaving B outside of the straws can only end up on B.

Now it is obvious that if the straws all end on the surface of each body without overlap, the area covered by the mouths of the straws on both bodies is the same. Since the bodies are the same size, the fraction of the surface that is covered is the same. Since black bodies radiate uniformly in all directions, the fraction of the rays going from A into the straws to B has to be the same as that going from B into the straws to A. Since both bodies start out at the same temperature, the number of rays and their frequencies are the same per unit time. Therefore, the bodies remain at the same temperature.

This proof also accounts for different sized bodies (the fraction of the radiation emitted into the straws by the two bodies is proportional to their areas) and bodies with different emissivities (the fraction of the radiation emitted into the straws is proportional to the ratio of their emissivities). These ratios are exactly what are needed to keep the temperature the same.

Congratulations to Bart, Arlin and Hans for resolving this very interesting paradox. The papaya battery is a dud. Some respondents have said that even if it worked it would not be efficient and would produce only small amounts of power. Perhaps. But anything that can produce energy absolutely free, can be recharged by the environment, and has no complex moving parts, would be huge. Once an optimum design was made with incremental engineering, it wouldn’t have been inconceivable to churn out these babies by the billions and place them additively in warm places. Talk about unlimited renewability! Unfortunately the thermodynamic policeman, the Law of Entropy, has decreed, as Qw’s poem nicely put it, that there ain’t no free lunch… So let me attempt the sequel:

The Second Law Strikes Back

Said the cop to the papaya:

“You’ve got points, Mr. P. but not any real substance.
Your blabbing might fool a lot of people, but I am not a dunce.

Once your points become objects, they quickly lose their focus,
Your energy difference vanishes, it was basically hocus-pocus.

Your objects can be big or small, it really doesn’t matter.
It will not change how warm they are, even if one is fatter.

By emitting radially, your problems aren’t alleviated,
‘Cuz from the exact radial line the atoms are deviated.

By the time they reach the other body, the rays are badly skewed,
They bounce right off again: the atoms can’t catch them, dude.

No clever trick, my brother, can skirt this iron Law:
If you can’t understand it, just think of the bendy straw!

So forget once for all, your grandiose dreams of cheating
Entropy will always increase, you’re sure to take a beating.

As long as I am around, it is pretty plain to see
You have to do good honest work, nothing comes for free.

So papaya, your future as a battery I’d say is pretty moot,
But you can give energy: just be who you are, man – A fruit!”

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John Tierney always wanted to be a scientist but went into journalism because its peer-review process was a great deal easier to sneak through. Now a columnist for the Science Times section, Tierney previously wrote columns for the Op-Ed page, the Metro section and the Times Magazine. Before that he covered science for magazines like Discover, Hippocrates and Science 86.

With your help, he's using TierneyLab to check out new research and rethink conventional wisdom about science and society. The Lab's work is guided by two founding principles:

Just because an idea appeals to a lot of people doesn't mean it's wrong.

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